# Properties

 Label 1600.2.s.b Level $1600$ Weight $2$ Character orbit 1600.s Analytic conductor $12.776$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.s (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 400) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{3} + ( 3 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + ( -1 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{3} + ( 3 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + ( -1 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{9} + ( -3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} ) q^{11} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{17} + ( 1 - 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{19} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{21} + ( -3 \zeta_{24} + \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{23} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{27} + ( 2 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{29} + ( -3 + 6 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{31} + ( -3 \zeta_{24} - 6 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{33} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 7 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{37} + ( 2 - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{39} + ( 2 - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{41} + ( \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{43} -2 \zeta_{24}^{3} q^{47} + ( -6 + 12 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{49} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{51} + ( 7 \zeta_{24} + 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{53} + ( -2 \zeta_{24} - 5 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{57} + ( 5 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{59} + ( -1 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{61} + ( -6 \zeta_{24} + 4 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{63} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{67} + ( -3 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{69} + ( 6 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{71} + ( -9 \zeta_{24} + 2 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{77} + ( 3 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{79} + ( -2 - 10 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{81} + ( -9 \zeta_{24}^{5} + 9 \zeta_{24}^{7} ) q^{83} + ( 2 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{87} + ( 12 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{89} + ( 12 - 12 \zeta_{24}^{2} - 12 \zeta_{24}^{4} + 12 \zeta_{24}^{6} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{93} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{97} + ( -3 - 3 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{9} + O(q^{10})$$ $$8q - 8q^{9} - 12q^{11} - 4q^{19} + 24q^{29} - 4q^{51} + 48q^{59} + 16q^{61} - 32q^{69} + 48q^{71} + 24q^{79} - 16q^{81} + 96q^{89} + 48q^{91} - 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-\zeta_{24}^{3}$$ $$-\zeta_{24}^{3}$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
207.1
 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i
0 −1.93185 0 0 0 −0.896575 0.896575i 0 0.732051 0
207.2 0 −0.517638 0 0 0 3.34607 + 3.34607i 0 −2.73205 0
207.3 0 0.517638 0 0 0 −3.34607 3.34607i 0 −2.73205 0
207.4 0 1.93185 0 0 0 0.896575 + 0.896575i 0 0.732051 0
943.1 0 −1.93185 0 0 0 −0.896575 + 0.896575i 0 0.732051 0
943.2 0 −0.517638 0 0 0 3.34607 3.34607i 0 −2.73205 0
943.3 0 0.517638 0 0 0 −3.34607 + 3.34607i 0 −2.73205 0
943.4 0 1.93185 0 0 0 0.896575 0.896575i 0 0.732051 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
80.j even 4 1 inner
80.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.s.b 8
4.b odd 2 1 400.2.s.b yes 8
5.b even 2 1 inner 1600.2.s.b 8
5.c odd 4 2 1600.2.j.b 8
16.e even 4 1 400.2.j.b 8
16.f odd 4 1 1600.2.j.b 8
20.d odd 2 1 400.2.s.b yes 8
20.e even 4 2 400.2.j.b 8
80.i odd 4 1 400.2.s.b yes 8
80.j even 4 1 inner 1600.2.s.b 8
80.k odd 4 1 1600.2.j.b 8
80.q even 4 1 400.2.j.b 8
80.s even 4 1 inner 1600.2.s.b 8
80.t odd 4 1 400.2.s.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.b 8 16.e even 4 1
400.2.j.b 8 20.e even 4 2
400.2.j.b 8 80.q even 4 1
400.2.s.b yes 8 4.b odd 2 1
400.2.s.b yes 8 20.d odd 2 1
400.2.s.b yes 8 80.i odd 4 1
400.2.s.b yes 8 80.t odd 4 1
1600.2.j.b 8 5.c odd 4 2
1600.2.j.b 8 16.f odd 4 1
1600.2.j.b 8 80.k odd 4 1
1600.2.s.b 8 1.a even 1 1 trivial
1600.2.s.b 8 5.b even 2 1 inner
1600.2.s.b 8 80.j even 4 1 inner
1600.2.s.b 8 80.s even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4 T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - 4 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$1296 + 504 T^{4} + T^{8}$$
$11$ $$( 81 - 54 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$13$ $$( 24 + T^{2} )^{4}$$
$17$ $$( 1 + T^{4} )^{2}$$
$19$ $$( 169 - 26 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$23$ $$456976 + 1784 T^{4} + T^{8}$$
$29$ $$( 144 - 144 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$31$ $$( 676 + 56 T^{2} + T^{4} )^{2}$$
$37$ $$( 4356 + 156 T^{2} + T^{4} )^{2}$$
$41$ $$( 9 + 42 T^{2} + T^{4} )^{2}$$
$43$ $$( 36 + 84 T^{2} + T^{4} )^{2}$$
$47$ $$( 16 + T^{4} )^{2}$$
$53$ $$( -98 + T^{2} )^{4}$$
$59$ $$( 4356 - 1584 T + 288 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$61$ $$( 2116 + 368 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$( 81 + 36 T^{2} + T^{4} )^{2}$$
$71$ $$( -12 - 12 T + T^{2} )^{4}$$
$73$ $$22667121 + 25074 T^{4} + T^{8}$$
$79$ $$( -18 - 6 T + T^{2} )^{4}$$
$83$ $$( 6561 - 324 T^{2} + T^{4} )^{2}$$
$89$ $$( 141 - 24 T + T^{2} )^{4}$$
$97$ $$26873856 + 72576 T^{4} + T^{8}$$